On the One-Dimensional Contact Process with Enhancements

TL;DR

The paper analyzes a one-dimensional contact process with two infection parameters, $\\lambda_i$ inside the infected region and $\\lambda_e$ at its boundary, establishing that the critical curves are strictly decreasing and continuous functions of the other parameter. It develops an enhanced graphical framework, speed analysis via the process seen from the edge, and block-renormalization techniques to show monotonicity and continuity of the critical values, as well as zero speed at criticality. The authors prove that survival can occur when one parameter is at the standard-critical value and the other is strictly larger, while also showing no survival for the finite-boundary-enhanced half-line case; they further derive domination results in non-attractive regimes and construct survival results for boundary-enhanced critical processes on the half-line. These results illuminate enhancement phenomena in oriented percolation-like systems and introduce methods—such as restart arguments and edge-invariant measures—that may apply to other interacting particle systems with boundary effects.

Abstract

We study a one-dimensional contact process with two infection parameters, one giving the infection rates at the boundaries of a finite infected region and the other one the rates within that region. We prove that the critical value of each of these parameters is a strictly monotone continuous function of the other parameter. We also show that if one of these parameters is equal to the critical value of the standard contact process and the other parameter is strictly larger, then the infection starting from a single point has positive probability of surviving. This is in contrast with another result also obtained here, that the critical contact process on the half line with enhanced infection rate at finitely many sites also dies out.

Figures (4)

• Figure 1.1: (a) Simple properties of the phase space: $\theta(\lambda_i,\lambda_e)>0$ if $\min\{\lambda_i,\lambda_e\}>\lambda_c$ and $\theta(\lambda_i,\lambda_e)=0$ if $\max\{\lambda_i,\lambda_e\} \leqslant \lambda_c$ or $\lambda_e \leqslant 1$. (b) Results of DurrettSchinazi00. (c) Results of this paper. (color online)
• Figure 5.1: Open branches in $D$ (left), $D'$ (center), and the grafting of $D'$ onto $D$ (right). Blue paths only jump to the right, and are $\lambda_e$-open. Red paths only jump to the left, and are $\lambda_e$-open. Green paths jump in both directions and are $\lambda_i$-open. For each open branch, at each instant in time, the rightmost site of the branch is occupied by a blue path and the leftmost site of the branch is occupied by a red path. Thin light gray lines indicate $t_0, t_0', t_1, t_1'$. A grayed version of $D$ behind $D'$ illustrates their relative position, and the four gray dots illustrate conditions $R(t_0') \leqslant L'(t_0')$, and $R'(t_1) \leqslant L(t_1)$. (color online)
• Figure 5.2: Illustration for $n=5$ of how simultaneous occurrence of $n+1$ similar events result in a complete open branch in $D_{4\ell,\frac{n}{2},\alpha}$. Green paths are $\lambda_i$-open, red and blue paths are $\lambda_e$-open. Existence of paths in the gray areas ensure that $\lambda_i$-open edges in between can be used. The green paths may be needed as they connect and lead to blue and red paths. (high resolution, color online)
• Figure 5.3: Good blocks and long paths.

Theorems & Definitions (64)

• theorem 1
• theorem 2
• Corollary 1.1
• theorem 3
• Corollary 1.2
• theorem 4
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Remark 1.6